This symposium is part of Project Palis-Balzan - Dynamical Systems, Chaotic Behaviour-Uncertainty, sponsored by the Balzan Foundation, from the award conferred to Jacob Palis by the Balzan Foundation in 2010 to Jacob Palis and IMPA and with support from CAPES, CNPq and FAPERJ. The aim of the project, involving scientists from different regions of the world and lasting for five years, has as main goal to advance the global conjecture of the finiteness of the number of attractors. Important topics of the project are also the linear cocycles and Lyapunov exponents. As participants of this project, we have Jacob Palis, and Jean-Christophe Yoccoz, Fields Medal in 1994, as coordinators and about 20 other renowned mathematicians from Brazil (Artur Avila, Carlos Gustavo Moreira, Enrique Pujals, Lorenzo Diaz, Marcelo Viana, Maria José Pacífico, Welington de Melo), and from France (Carlos Matheus, Sylvain Crovisier, Christian Bonatti and Pierre Berger), U.S. (Marco Martens, Michael Lyubich), England (Sebastian van Strien), China (Lan Wen) and Uruguay (Martin Sambarino, Alvaro Rovella), and about 10 young doctors. This symposium aims to promote research at the highest level in the area of dynamical systems, especially in the above topics, with the effective participation of excellent research groups of foreign and national researchers. It also aims to put post-doctoral students and young researchers in contact with the best that is produced worldwide in the above and related issues, disseminating recent results and providing a scientific exchange on a world level. In particular, this encourages the development of the Brazilian group in the area, which is increasingly asserting itself in the international arena.